I am wondering what currently our mathematicians do related to conformal field theory, (I know currently it is a central topic, but I have only a vague idea what mathematicians do in there), or more generally topological field theory, field theory... I am extremely appreicated if there is any survey paper.

5 Answers
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Costello, "Topological conformal field theories and Calabi-Yau categories" -- This is (essentially) the 2d version of the (Hopkins-)Lurie/Baez-Dolan cobordism hypothesis that Lennart mentions. See also Kontsevich-Soibelman, "Notes on A-infinity...". This stuff is closely related to mirror symmetry, which is - in physics terms - a duality between certain field theories (or sigma models). Mirror symmetry by itself is already a huge enterprise...

See papers by Yi-Zhi Huang for stuff about vertex operator algebras and CFTs.

One can consider string topology from a field theory viewpoint... see for example Sullivan, "String Topology: Background and Present State" and Blumberg-Cohen-Teleman, "Open-closed field theories, string topology, and Hochschild homology". This is actually related to the work of Costello, Lurie, Kontsevich mentioned above -- see e.g. section 2.1 of Costello's paper.

An important problem is that of making rigorous some of the things that physicists do in QFT, such as path integrals. See Costello, "Renormalization and effective field theory" and also Borcherds, "Renormalization and quantum field theory".

One direction is the attempt of Igor Kriz to make conformal field theory rigorous in terms of lax algebras and elliptic cohomology. This work also addresses the interesting problem of understanding the moduli space of rational conformal field theories, a problem that's also important in string theory, where certain types of rational conformal field theories can be related to Calabi-Yau varieties. The deformations of the CFT then are to be mapped to the deformations of the Calabi-Yau families. This raises interesting problems because of the singularities encountered in the CY moduli space. Papers in this direction can be found on his webpage at Michigan.

More generally, there is much work done by Bloch, Marcolli and others to understand the appearance of motives in perturbative quantum field theory. Papers in this direction can be found on Bloch's Chicago webpage and Marcolli's Caltech webpage. Marcolli has also recently written a book on this theme called "Feynman Motives".

Note that the word "conformal field theory" (indeed, "field theory" in general) has many meanings, depending on the area of mathematics of the user, and in general the relationships between the different meanings is only conjectural. For one use of the word, with applications to the representation theory of loop groups, you might check out the notes form the recent Workshop on Operator Algebras and Conformal Field Theory at University of Oregon.

It is a huge topic (or a more than one huge topic) and I am not an expert, but I want to offer two examples of what is currently en vogue in topological field theories:

1) Extended Field Theories: Classicly you give for every n-dimensional a vector space and for every (n+1)-dimensional bordism a morphism. Extended can mean that you replace the category of vector spaces by a suitable 2-category and give for every bordism between bordisms a 2-morphism. For more precise statements and a fancy classification result, you should have a look at: http://www.math.harvard.edu/~lurie/papers/cobordism.pdf

2) Another important topic is the connection between 2-dimensional conformal field theories and 3-dimension topological field theories - or more generally: how does one construct 3-dimensional topological field theories? A good introduction is the book of Bakalov and Kirillov - Lectures on Tensor Categories and Modular Functors.